The Exact Theory of the Stern–Gerlach Experiment and Why It Does Not Imply That a Fermion Can Only Have Its Spin Up or Down

• Main Author: Gerrit Coddens
• Format: Article
• Language: English
• Published: MDPI AG 2021-01-01
• Series: Symmetry
• Subjects: quantum mechanics; SU(2); Stern–Gerlach experiment; spinors
• Online Access: https://www.mdpi.com/2073-8994/13/1/134

The Stern–Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern–Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles <inline-formula><math display="inline"><semantics><mi>θ</mi></semantics></math></inline-formula> between the spin axis <inline-formula><math display="inline"><semantics><mi mathvariant="bold">s</mi></semantics></math></inline-formula> and the magnetic field <inline-formula><math display="inline"><semantics><mi mathvariant="bold">B</mi></semantics></math></inline-formula>, the fermions can only have two possible energies <inline-formula><math display="inline"><semantics><mrow><msub><mi>m</mi><mn>0</mn></msub><msup><mi>c</mi><mn>2</mn></msup><mo>±</mo><mi>μ</mi><mi>B</mi></mrow></semantics></math></inline-formula>. The values <inline-formula><math display="inline"><semantics><mrow><mo>±</mo><mi>μ</mi><mi>B</mi></mrow></semantics></math></inline-formula> thus do not correspond to the continuum of values <inline-formula><math display="inline"><semantics><mrow><mo>−</mo><mrow><mi>μ</mi><mo>·</mo></mrow><mi mathvariant="bold">B</mi></mrow></semantics></math></inline-formula> Einstein and Ehrenfest had conjectured. The energy term <inline-formula><math display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mo>−</mo><mrow><mi>μ</mi><mo>·</mo></mrow><mi mathvariant="bold">B</mi></mrow></semantics></math></inline-formula> is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies <inline-formula><math display="inline"><semantics><mrow><mo>±</mo><mi>μ</mi><mi>B</mi></mrow></semantics></math></inline-formula>, and as such not valid for individual fermions. The two fermion states with energy <inline-formula><math display="inline"><semantics><mrow><mo>±</mo><mi>μ</mi><mi>B</mi></mrow></semantics></math></inline-formula> are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.